Cartesian Product is Small

Theorem

Let $a$ and $b$ be small classes.

Then their Cartesian product $a \times b$ is small:

$\map {\MM} {a \times b}$

Proof

By Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union:

$a \times b \subseteq \powerset {\powerset {a \cup b} }$

By Union of Small Classes is Small, $a \cup b$ is small.

By the Axiom of Powers, $\powerset {\powerset {a \cup b} }$ is small.

By Axiom of Subsets Equivalents, $a \times b$ is small.

$\blacksquare$

Sources

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• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.2$